Braids, normed division algebras, and Standard Model symmetries

Abstract

This paper represents a first attempt at unifying two promising models that attempt to explain the origin of the internal symmetries of leptons and quarks. It is shown that each of the four normed division algebras over the reals admits a representation of a circular braid group. For the complex numbers and the quaternions, the represented circular braid groups are $B_2$ and $B_3^c$, precisely those used to construct leptons and quarks as framed braids in the Helon model of Bilson-Thompson. It is then shown that these framed braids coincide with the states that span the minimal left ideals of the complex (chained) octonions, shown by Furey to describe one generation of leptons and quarks with unbroken $SU(3{)}_c$ and $U(1{)}_{em}$ symmetry. The identification of basis states of minimal ideals with certain framed braids is possible because the braiding in $B_2$ and $B_3^c$ in the Helon model are interchangeable. It is shown that the framed braids in the Helon model can be written as pure braid words in $B_3^c$ with trivial braiding in $B_2$, something which is not possible for framed braids in general.

20 pages, 4 figures

References in corpus (10)

• Quantum gravity and the standard model
• Generations: Three Prints, in Colour
• A topological model of composite preons
• Standard model physics from an algebra?
• Propagation and interaction of chiral states in quantum gravity
• Particle Identifications from Symmetries of Braided Ribbon Network Invariants
• C, P, and T of Braid Excitations in Quantum Gravity
• The Stabilized Poincare-Heisenberg algebra: a Clifford algebra viewpoint
• Effective Theory of Braid Excitations of Quantum Geometry in terms of Feynman Diagrams
• Quantum groups and braid groups as fundamental symmetries

Cited by in corpus (18)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• Division algebraic symmetry breaking
• The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
• An Exceptional G(2) Extension of the Standard Model from the Correspondence with Cayley-Dickson Algebras Automorphism Groups
• Braids, 3-manifolds, elementary particles: number theory and symmetry in particle physics
• A topological model of composite preons from the minimal ideals of two Clifford algebras
• The $C\ell (8)$ algebra of three fermion generations with spin and full internal symmetries
• Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions
• Topological Quantum Computing and 3-Manifolds
• Braided fermions from Hurwitz algebras
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Braided matter interactions in quantum gravity via 1-handle attachment
• Preons, Braid Topology, and Representations of Fundamental Particles
• Knotted boundaries and braid only form of braided belts
• An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints
• An Algebraic Roadmap of Particle Theories, Part I: General construction